Problem 58 - Essential Calculus

Question

Use a graphing utility to graph the polar equation and find all points of horizontal tangency. $$ r=3 \cos 2 \theta \sec \theta $$

Step-by-Step Solution

Answer

The complete solution involves the conversion of the polar equation to Cartesian coordinates, the graphing of this function and finally to find where the derivative wrt \( y \) equals zero. This will provide the points of horizontal tangency.

Step 1: Conversion to Cartesian Coordinates

To graph the function using a graphing utility, we need to convert the polar equation into Cartesian coordinates. Remember that we may express \( r = x^2 + y^2 \) and \( \theta = \arctan(\frac{y}{x}) \). Here, multiply \( 3 \cos 2 \theta \sec \theta = 3 \cos 2 \theta / \cos \theta \) by \( \sqrt{x^2 + y^2} \) to get in terms of 'x' and 'y'. After simplification, you will obtain an equation in terms of 'x' and 'y'.

Step 2: Graphing the function

Use any graphing utility to plot the Cartesian equation you obtained from step 1. For each point, you can then convert the Cartesian coordinates back to polar coordinates.

Step 3: Determining the points of horizontal tangency

A point of horizontal tangency will exist where the derivative of the equation wrt \( y \) equals zero, as the tangent line will be horizontal to the x-axis at these points. Therefore, find the derivative of the equation wrt \( y \), set it equal to zero, and solve to find these points.